Let $K$ be any field. If $A$ is the zero set of a polynomial $P\in K[X]$, then $A$ is finite. This follows from the fact that $K[X]$ is Euclidian, using commutativity of $K$.
Now let $A\subset K^n$ be an algebraic subset, i.e. the zero set of finitely many polynomials $P_1,\dots,P_q\in K[X_1,\dots,X_n]$.
If all points of $A$ are isolated for the Zariski topology on $K^n$, i.e. $A$ is discrete, why is $A$ necessarilly finite ?
$A$ is quasi-compact, since it is a closed subset of the quasi-compact set $K^n$. Any quasi-compact discrete set is finite, this is a very easy exercise in basic topology.